Ill-posedness for the stationary Navier-Stokes equations in critical Besov spaces

Abstract

This paper presents some progress toward an open question which proposed by Tsurumi (Arch. Ration. Mech. Anal. 234:2, 2019): whether or not the stationary Navier-Stokes equations in ${\mathbb{R}}^d$ is well-posed from ${\dot{B}}_{p,q}^{-2}$ to $\mathbb{P}{\dot{B}}_{p,q}^0$ with $p=d$ and $1\le q\le 2$. In this paper, we prove that for the case $1\le q<\frac{d}{2}$ with $d\ge 4$ the stationary Navier-Stokes equations is ill-posed from ${\dot{B}}_{d,q}^{-2}({\mathbb{R}}^d)$ to $\mathbb{P}{\dot{B}}_{d,q}^0({\mathbb{R}}^d)$ by showing that a sequence of external forces is constructed to show discontinuity of the solution map at zero. Indeed in such case of $q$, there exists a sequence of external forces which converges to zero in ${\dot{B}}_{d,q}^{-2}$ and yields a sequence of solutions which does not converge to zero in ${\dot{B}}_{d,q}^0$. In particular, we also prove that the stationary Navier-Stokes equations is well-posed from ${\dot{B}}_{d,2}^{-2}({\mathbb{R}}^d)$ to $\mathbb{P}{\dot{B}}_{d,2}^0({\mathbb{R}}^d)$ with $d=3,4$. Based on these two cases, we demonstrate that the above open question for the dimension $d\ge 4$ has been solved completely.